Math212a1010 Lebesgue measure.

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Math212a1010 Lebesgue measure."

Transcription

1 Math212a1010 Lebesgue measure. October 19, 2010

2 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.

3 Henri Léon Lebesgue Born: 28 June 1875 in Beauvais, Oise, Picardie, France Died: 26 July 1941 in Paris, France

4 In today s lecture we will discuss the concept of measurability of a subset of R. We will begin with Lebesgue s (1902) definition of measurability, which is easy to understand and intuitive. We will then give Caratheodory s (1914) definition of measurabiity which is highly non-intuitive but has great technical advantage. For subsets of R these two definitions are equivalent (as we shall prove). But the Caratheodory definition extends to many much more general situations. In particular, the Caratheodory definition will prove useful for us later, when we study Hausdorff measures.

5 The ɛ/2 n trick. An argument which will recur several times is: and so n = 1 ɛ 2 n = ɛ. We will call this the ɛ/2 n trick and not spell it out every time we use it.

6 1 Lebesgue outer measure. 2 Lebesgue inner measure. 3 Lebesgue s definition of measurability. 4 Caratheodory s definition of measurability. 5 Countable additivity. 6 σ-fields, measures, and outer measures.

7 The definition of Lebesgue outer measure. For any subset A R we define its Lebesgue outer measure by m (A) := inf l(i n ) : I n are intervals with A I n. (1) Here the length l(i ) of any interval I = [a, b] is b a with the same definition for half open intervals (a, b] or [a, b), or open intervals. Of course if a = and b is finite or +, or if a is finite and b = +, or if a = and b = the length is infinite.

8 It doesn t matter if the intervals are open, half open or closed. m (A) := inf l(i n ) : I n are intervals with A I n. (1) The infimum in (1) is taken over all covers of A by intervals. By the ɛ/2 n trick, i.e. by replacing each I j = [a j, b j ] by (a j ɛ/2 j+1, b j + ɛ/2 j+1 ) we may assume that the infimum is taken over open intervals. (Equally well, we could use half open intervals of the form [a, b), for example.).

9 Monotonicity, subadditivity. If A B then m (A) m (B) since any cover of B by intervals is a cover of A. For any two sets A and B we clearly have m (A B) m (A) + m (B).

10 Sets of measure zero don t matter. A set Z is said to be of (Lebesgue) measure zero it its Lebesgue outer measure is zero, i.e. if it can be covered by a countable union of (open) intervals whose total length can be made as small as we like. If Z is any set of measure zero, then m (A Z) = m (A).

11 The outer measure of a finite interval is its length. If A = [a, b] is an interval, then we can cover it by itself, so m ([a, b]) b a, and hence the same is true for (a, b], [a, b), or (a, b). If the interval is infinite, it clearly can not be covered by a set of intervals whose total length is finite, since if we lined them up with end points touching they could not cover an infinite interval. We claim that if I is a finite interval. m (I ) = l(i ) (2)

12 Proof. We may assume that I = [c, d] is a closed interval by what we have already said, and that the minimization in (1) is with respect to a cover by open intervals. So what we must show is that if [c, d] i (a i, b i ) then d c i (b i a i ). We first apply Heine-Borel to replace the countable cover by a finite cover. (This only decreases the right hand side of preceding inequality.) So let n be the number of elements in the cover.

13 We need to prove that if n [c, d] (a i, b i ) then d c i=1 n (b i a i ). i=1 We do this this by induction on n. If n = 1 then a 1 < c and b 1 > d so clearly b 1 a 1 > d c. Suppose that n 2 and we know the result for all covers (of all intervals [c, d] ) with at most n 1 intervals in the cover. If some interval (a i, b i ) is disjoint from [c, d] we may eliminate it from the cover, and then we are in the case of n 1 intervals. So we may assume that every (a i, b i ) has non-empty intersection with [c, d].

14 Among the the intervals (a i, b i ) there will be one for which a i takes on the minimum possible value. By relabeling, we may assume that this is (a 1, b 1 ). Since c is covered, we must have a 1 < c. If b 1 > d then (a 1, b 1 ) covers [c, d] and there is nothing further to do. So assume b 1 d. We must have b 1 > c since (a 1, b 1 ) [c, d]. Since b 1 [c, d], at least one of the intervals (a i, b i ), i > 1 contains the point b 1. By relabeling, we may assume that it is (a 2, b 2 ). But now we have a cover of [c, d] by n 1 intervals: [c, d] (a 1, b 2 ) n (a i, b i ). i=3 So by induction d c (b 2 a 1 ) + n i=3 (b i a i ). But b 2 a 1 (b 2 a 2 ) + (b 1 a 1 ) since a 2 < b 1.

15 We can use small intervals. We repeat that the intervals used in (1) could be taken as open, closed or half open without changing the definition. If we take them all to be half open, of the form I i = [a i, b i ), we can write each I i as a disjoint union of finite or countably many intervals each of length < ɛ. So it makes no difference to the definition if we also require the l(i i ) < ɛ (3) in (1). We will see that when we pass to other types of measures this will make a difference.

16 Summary of where we are so far. We have verified, or can easily verify the following properties: 1 m ( ) = 0. 2 A B m (A) m (B). 3 By the ɛ/2 n trick, for any finite or countable union we have m ( i A i ) i m (A i ). 4 If dist (A, B) > 0 then m (A B) = m (A) + m (B). 5 m (A) = inf{m (U) : U A, U open}. 6 For an interval m (I ) = l(i ).

17 The only items that we have not done already are items 4 and 5. But these are immediate: for 4 we may choose the intervals in (1) all to have length < ɛ where 2ɛ < dist (A, B) so that there is no overlap. As for item 5, we know from 2 that m (A) m (U) for any set U A, in particular for any open set U which contains A. We must prove the reverse inequality: if m (A) = this is trivial. Otherwise, we may take the intervals in (1) to be open and then the union on the right is an open set whose Lebesgue outer measure is less than m (A) + δ for any δ > 0 if we choose a close enough approximation to the infimum.

18 Lebesgue outer measure on R n. All the above works for R n instead of R if we replace the word interval by rectangle, meaning a rectangular parallelepiped, i.e a set which is a product of one dimensional intervals. We also replace length by volume (or area in two dimensions). What is needed is the following:

19 Lemma Let C be a finite non-overlapping collection of closed rectangles all contained in the closed rectangle J. Then vol J I C vol I. If C is any finite collection of rectangles such that J I C I then vol J I C vol (I ).

20 This lemma occurs on page 1 of Stroock, A concise introduction to the theory of integration together with its proof. I will take this for granted. In the next few slides I will talk as if we are in R, but everything goes through unchanged if R is replaced by R n. In fact, once we will have developed enough abstract theory, we will not need this lemma. So for those of you who are purists and do not want to go through the lemma in Stroock, just stick to R.

21 The definition of Lebesgue inner measure. Item 5. in our list said that the Lebesgue outer measure of any set is obtained by approximating it from the outside by open sets. The Lebesgue inner measure is defined as m (A) = sup{m (K) : K A, K compact }. (4) Clearly m (A) m (A) since m (K) m (A) for any K A. We also have

22 The Lebesgue inner measure of an interval. Proposition. For any interval I we have m (I ) = l(i ). (5)

23 Proof. If l(i ) = the result is obvious. So we may assume that I is a finite interval which we may assume to be open, I = (a, b). If K I is compact, then I is a cover of K and hence from the definition of outer measure m (K) l(i ). So m (I ) l(i ). On the other hand, for any ɛ > 0, ɛ < 1 2 (b a) the interval [a + ɛ, b ɛ] is compact and m ([a ɛ, a + ɛ]) = b a 2ɛ m (I ). Letting ɛ 0 proves the proposition.

24 Lebesgue s definition of measurability for sets of finite outer measure. A set A with m (A) < is said to measurable in the sense of Lebesgue if m (A) = m (A). (6) If A is measurable in the sense of Lebesgue, we write m(a) = m (A) = m (A). (7) If K is a compact set, then m (K) = m (K) since K is a compact set contained in itself. Hence all compact sets are measurable in the sense of Lebesgue. If I is a bounded interval, then I is measurable in the sense of Lebesgue by the preceding Proposition.

25 Lebesgue s definition of measurability for sets of infinite outer measure. If m (A) =, we say that A is measurable in the sense of Lebesgue if all of the sets A [ n, n] are measurable.

26 Countable additivity. Theorem If A = A i is a (finite or) countable disjoint union of sets which are measurable in the sense of Lebesgue, then A is measurable in the sense of Lebesgue and m(a) = i m(a i ). In the proof we may assume that m(a) < - otherwise apply the result to A [ n, n] and A i [ n, n] for each n.

27 We have m (A) n m (A n ) = n m(a n ). Let ɛ > 0, and for each n choose compact K n A n with m (K n ) m (A n ) ɛ 2 n = m(a n) ɛ 2 n which we can do since A n is measurable in the sense of Lebesgue. The sets K n are pairwise disjoint, hence, being compact, at positive distances from one another. Hence m (K 1 K n ) = m (K 1 ) + + m (K n ) and K 1 K n is compact and contained in A.

28 m (K n ) m (A n ) ɛ 2 n = m(a n) ɛ 2 n m (K 1 K n ) = m (K 1 ) + + m (K n ) and K 1 K n is compact and contained in A. Hence m (A) m (K 1 ) + + m (K n ), and since this is true for all n we have m (A) n m(a n ) ɛ. Since this is true for all ɛ > 0 we get m (A) m(a n ). But then m (A) m (A) and so they are equal, so A is measurable in the sense of Lebesgue, and m(a) = m(a i ).

29 Open sets are measurable in the sense of Lebesgue. Proof. Any open set O can be written as the countable union of open intervals I i, and n 1 J n := I n \ is a disjoint collection of intervals (some open, some closed, some half open) and O is the disjont union of the J n. So every open set is a disjoint union of intervals hence measurable in the sense of Lebesgue. i=1 I i

30 Closed sets are measurable in the sense of Lebesgue. This requires a bit more work: If F is closed, and m (F ) =, then F [ n, n] is compact, and so F is measurable in the sense of Lebesgue. Suppose that m (F ) <.

31 For any ɛ > 0 consider the sets G 1,ɛ := [ 1 + ɛ 2 2, 1 ɛ 2 2 ] F G 2,ɛ := ([ 2 + ɛ ɛ, 1] F ) ([1, ] F ) G 3,ɛ := ([ 3 + ɛ ɛ, 2] F ) ([2, ] F ). and set G ɛ := i G i,ɛ. The G i,ɛ are all compact, and hence measurable in the sense of Lebesgue, and the union in the definition of G ɛ is disjoint, so G ɛ is measurable in the sense of Lebesgue.

32 The G i,ɛ are all compact, and hence measurable in the sense of Lebesgue, and the union in the definition of G ɛ is disjoint, so is measurable in the sense of Lebesgue. Furthermore, the sum of the lengths of the gaps between the intervals that went into the definition of the G i,ɛ is ɛ. So m(g ɛ )+ɛ = m (G ɛ )+ɛ m (F ) m (G ɛ ) = m(g ɛ ) = i m(g i,ɛ ). In particular, the sum on the right converges, and hence by considering a finite number of terms, we will have a finite sum whose value is at least m(g ɛ ) ɛ. The corresponding union of sets will be a compact set K ɛ contained in F with m(k ɛ ) m (F ) 2ɛ. Hence all closed sets are measurable in the sense of Lebesgue.

33 The inner-outer characterization of measurability. Theorem A is measurable in the sense of Lebesgue if and only if for every ɛ > 0 there is an open set U A and a closed set F A such that m(u \ F ) < ɛ. Proof in one direction. Suppose that A is measurable in the sense of Lebesgue with m(a) <. Then there is an open set U A with m(u) < m (A) + ɛ/2 = m(a) + ɛ/2, and there is a compact set F A with m(f ) m (A) ɛ = m(a) ɛ/2. Since U \ F is open, it is measurable in the sense of Lebesgue, and so is F as it is compact. Also F and U \ F are disjoint. Hence m(u \ F ) = m(u) m(f ) < m(a) + ɛ ( 2 m(a) ɛ ) = ɛ. 2

34 If A is measurable in the sense of Lebesgue, and m(a) =, we can apply the above to A I where I is any compact interval. So we can find open sets U n A [ n 2δ n+1, n + 2δ n+1 ] and closed sets F n A [ n, n] with m(u n \ F n ) < ɛ/2 n. Here the δ n are sufficiently small positive numbers. We may enlarge each F n if necessary so that F n [ n + 1, n 1] F n 1.

35 We may also decrease the U n if necessary so that U n ( n + 1 δ n, n 1 + δ n ) U n 1. Indeed, if we set C n := [ n + 1 δ n, n 1 + δ n ] U c n 1 then C n is a closed set with C n A =. Then U n C c n is still an open set containing [ n 2δ n+1, n + 2δ n+1 ] A and (U n C c n ) ( n+1 δ n, n 1+δ n ) C c n ( n+1 δ n, n 1+δ n ) U n 1.

36 Take U := U n so U is open. Take F := (F n ([ n, n + 1] [n 1, n])). Then F is closed, U A F and U \ F (U n /F n ) ([ n, n + 1] [n 1, n]) (U n \ F n ) this concludes the proof in one direction.

37 Proof in the other direction. Suppose that for each ɛ, there exist U A F with m(u \ F ) < ɛ. Suppose that m (A) <. Then m(f ) < and m(u) m(u \ F ) + m(f ) < ɛ + m(f ) <. Then m (A) m(u) < m(f ) + ɛ = m (F ) + ɛ m (A) + ɛ. Since this is true for every ɛ > 0 we conclude that m (A) m (A) so they are equal and A is measurable in the sense of Lebesgue. If m (A) =, we have U ( n ɛ, n + ɛ) A [ n, n] F [ n, n] and m((u ( n ɛ, n + ɛ) \ (F [ n, n]) < 2ɛ + ɛ = 3ɛ so we can proceed as before to conclude that m (A [ n, n]) = m (A [ n, n]).

38 Consequences of the theorem. Proposition. If A is measurable in the sense of Lebesgue, so is its complement A c = R \ A. Proof. Indeed, if F A U with F closed and U open, then F c A c U c with F c open and U c closed. Furthermore, F c \ U c = U \ F so if A satisfies the condition of the theorem so does A c.

39 More consequences. Proposiition. If A and B are measurable in the sense of Lebesgue so is A B Proof. For ɛ > 0 choose U A A F A and U B B F B with m(u A \ F A ) < ɛ/2 and m(u B \ F B ) < ɛ/2. Then (U A U B ) (A B) (F A F B ) and (U A U B ) \ (F A F B ) (U A \ F A ) (U B \ F B ).

40 Still more consequences of the theorem. Proposition. If A and B are measurable in the sense of Lebesgue then so is A B. Proof. Indeed, A B = (A c B c ) c. Since A \ B = A B c we also get Proposition. If A and B are measurable in the sense of Lebesgue then so is A \ B.

41 Caratheodory s definition of measurability. A set E R is said to be measurable according to Caratheodory if for any set A R we have m (A) = m (A E) + m (A E c ) (8) where (recall that) E c denotes the complement of E. In other words, A E c = A \ E. This definition has many advantages, as we shall see. Our first task will be to show that it is equivalent to Lebesgue s. Notice that we always have m (A) m (A E) + m (A \ E) so condition (8) is equivalent to for all A. m (A E) + m (A \ E) m (A) (9)

42 Caratheodory = Lebesgue. Theorem A set E is measurable in the sense of Caratheodory if and only if it is measurable in the sense of Lebesgue. Proof that Lebesgue Caratheodory. Suppose E is measurable in the sense of Lebesgue. Let ɛ > 0. Choose U E F with U open, F closed and m(u/f ) < ɛ which we can do by the inner-outer Theorem. Let V be an open set containing A. Then A \ E V \ F and A E (V U) so

43 A \ E V \ F and A E (V U) m (A \ E) + m (A E) m(v \ F ) + m(v U) so m(v \ U) + m(u \ F ) + m(v U) m(v ) + ɛ. (We can pass from the second line to the third since both V \ U and V U are measurable in the sense of Lebesgue and we can apply the proposition about disjoint unions.) Taking the infimum over all open V containing A, the last term becomes m (A) + ɛ, and as ɛ is arbitrary, we have established (9) showing that E is measurable in the sense of Caratheodory.

44 Caratheodory Lebesgue, case where m (E) <. Then for any ɛ > 0 there exists an open set U E with m(u) < m (E) + ɛ. We may apply condition (8) to A = U to get so m(u) = m (U E) + m (U \ E) = m (E) + m (U \ E) m (U \ E) < ɛ. This means that there is an open set V (U \ E) with m(v ) < ɛ. But we know that U \ V is measurable in the sense of Lebesgue, since U and V are, and m(u) m(v ) + m(u \ V ) so m(u \ V ) > m(u) ɛ.

45 m(u \ V ) > m(u) ɛ. So there is a closed set F U \ V with m(f ) > m(u) ɛ. But since V U \ E = U E c, we have U \ V = U V c U (U c E) = E. So F E. So F E U and m(u \ F ) = m(u) m(f ) < ɛ. Hence E is measurable in the sense of Lebesgue.

46 Caratheodory Lebesgue, case where m (E) =. If m(e) =, we must show that E [ n, n] is measurable in the sense of Caratheodory, for then it is measurable in the sense of Lebesgue from what we already know. We know that the interval [ n, n] itself, being measurable in the sense of Lebesgue, is measurable in the sense of Caratheodory. So we will have completed the proof of the theorem if we show that the intersection of E with [ n, n] is measurable in the sense of Caratheodory.

47 Unions and intersections. More generally, we will show that the union or intersection of two sets which are measurable in the sense of Caratheodory is again measurable in the sense of Caratheodory. Notice that the definition (8) is symmetric in E and E c so if E is measurable in the sense of Caratheodory so is E c. So it suffices to prove the next lemma to complete the proof. Lemma If E 1 and E 2 are measurable in the sense of Caratheodory so is E 1 E 2.

48 Proof of the lemma, 1. For any set A we have m (A) = m (A E 1 ) + m (A E1 c ) by (8) applied to E 1. Applying (8) to A E1 c and E 2 gives m (A E1 c ) = m (A E1 c E 2 ) + m (A E1 c E2 c ). Substituting this back into the preceding equation gives m (A) = m (A E 1 ) + m (A E1 c E 2 ) + m (A E1 c E2 c ). (10) Since E1 c E 2 c = (E 1 E 2 ) c we can write this as m (A) = m (A E 1 ) + m (A E1 c E 2 ) + m (A (E 1 E 2 ) c ).

49 Proof of the lemma, 2. m (A) = m (A E 1 ) + m (A E c 1 E 2 ) + m (A (E 1 E 2 ) c ). Now A (E 1 E 2 ) = (A E 1 ) (A (E c 1 E 2) so m (A E 1 ) + m (A E c 1 E 2 ) m (A (E 1 E 2 )). Substituting this for the two terms on the right of the previous displayed equation gives m (A) m (A (E 1 E 2 )) + m (A (E 1 E 2 ) c ) which is just (9) for the set E 1 E 2. This proves the lemma and the theorem.

50 The class M of measurable sets. We let M denote the class of measurable subsets of R - measurability in the sense of Lebesgue or Caratheodory these being equivalent. Notice by induction starting with two terms as in the lemma, that any finite union of sets in M is again in M

51 The first main theorem in the subject is the following description of M and the function m on it: Theorem M and the function m : M [0, ] have the following properties: R M. E M E c M. If E n M for n = 1, 2, 3,... then n E n M. If F n M and the F n are pairwise disjoint, then F := n F n M and m(f ) = m(f n ). n=1

52 We already know the first two items on the list, and we know that a finite union of sets in M is again in M. We also know the last assertion. But it will be instructive and useful for us to have a proof starting directly from Caratheodory s definition of measurablity:

53 Recall m (A) = m (A E 1 )+m (A E c 1 E 2 )+m (A E c 1 E c 2 ). (10) If F 1 M, F 2 M and F 1 F 2 = then taking A = F 1 F 2, E 1 = F 1, E 2 = F 2 in (10) gives m(f 1 F 2 ) = m(f 1 ) + m(f 2 ). Induction then shows that if F 1,..., F n are pairwise disjoint elements of M then their union belongs to M and m(f 1 F 2 F n ) = m(f 1 ) + m(f 2 ) + + m(f n ).

54 m (A) = m (A E 1 )+m (A E c 1 E 2 )+m (A E c 1 E c 2 ). (10) More generally, if we let A be arbitrary and take E 1 = F 1, E 2 = F 2 in (10) we get m (A) = m (A F 1 ) + m (A F 2 ) + m (A (F 1 F 2 ) c ). If F 3 M is disjoint from F 1 and F 2 we may apply (8) with A replaced by A (F 1 F 2 ) c and E by F 3 to get m (A (F 1 F 2 ) c )) = m (A F 3 ) + m (A (F 1 F 2 F 3 ) c ), since (F 1 F 2 ) c F c 3 = F c 1 F c 2 F c 3 = (F 1 F 2 F 3 ) c. Substituting this back into the preceding equation gives m (A) = m (A F 1 )+m (A F 2 )+m (A F 3 )+m (A (F 1 F 2 F 3 ) c ).

55 m (A) = m (A F 1 )+m (A F 2 )+m (A F 3 )+m (A (F 1 F 2 F 3 ) c ). Proceeding inductively, we conclude that if F 1,..., F n are pairwise disjoint elements of M then m (A) = n m (A F i ) + m (A (F 1 F n ) c ). (11) 1

56 Now suppose that we have a countable family {F i } of pairwise disjoint sets belonging to M. Since ( n ) c ( ) c F i F i i=1 i=1 we conclude from (11) that ( n ) c ) m (A) m (A F i ) + m (A F i 1 i=1 and hence passing to the limit ( ) c ) m (A) m (A F i ) + m (A F i. 1 i=1

57 Now given any collection of sets B k we can find intervals {I k,j } with B k I k,j j and m (B k ) j l(i k,j ) + ɛ 2 k. So and hence B k k k,j I k,j m ( Bk ) m (B k ), the inequality being trivially true if the sum on the right is infinite.

58 So i=1 m (A F k ) m (A ( i=1 F i)). Thus ( ) c ) m (A) m (A F i ) + m (A F i 1 i=1 ( )) ( ) c ) m (A F i + m (A F i. i=1 The extreme right of this inequality is the left hand side of (9) applied to E = F i, i and so E M and the preceding string of inequalities must be equalities since the middle is trapped between both sides which must be equal. i=1

59 Hence we have proved that if F n is a disjoint countable family of sets belonging to M then their union belongs to M and m (A) = ( ) c ) m (A F i ) + m (A F i. (12) i i=1 If we take A = F i we conclude that if the F j are disjoint and m(f ) = m(f n ) (13) n=1 F = F j. So we have reproved the last assertion of the theorem using Caratheodory s definition.

60 Proof of 3:If E n M for n = 1, 2, 3,... then n E n M. For the proof of this third assertion, we need only observe that a countable union of sets in M can be always written as a countable disjoint union of sets in M. Indeed, set F 1 := E 1, F 2 := E 2 \ E 1 = E 1 E c 2 F 3 := E 3 \ (E 1 E 2 ) etc. The right hand sides all belong to M since M is closed under taking complements and finite unions and hence intersections, and F j = E j. We have completed the proof of the theorem. j

61 Some consequences - symmetric differences. The symmetric difference between two sets is the set of points belonging to one or the other but not both: A B := (A \ B) (B \ A). Proposition. If A M and m(a B) = 0 then B M and m(a) = m(b).

62 Proof. By assumption A \ B has measure zero (and hence is measurable) since it is contained in the set A B which is assumed to have measure zero. Similarly for B \ A. Also (A B) M since A B = A \ (A \ B). Thus B = (A B) (B \ A) M. Since B \ A and A B are disjoint, we have m(b) = m(a B)+m(B\A) = m(a B) = m(a B)+m(A\B) = m(a).

63 More consequences: increasing limits. Proposition. Suppose that A n M and A n A n+1 for n = 1, 2,.... Then ( ) m An = lim m(a n). n

64 Proof. Setting B n := A n \ A n 1 (with B 1 = A 1 ) the B i are pairwise disjoint and have the same union as the A i so ( ) m An = = lim n m i=1 ( n i=1 m(b i ) = lim n n m(b n ) i=1 B i ) = lim n m(a n).

65 More consequences: decreasing limits. Proposition. If C n C n+1 is a decreasing family of sets in M and m(c 1 ) < then ( ) m Cn = lim m(c n). n

66 Proof. Set A 1 :=, A 2 := C 1 \ C 2, A 3 := C 1 \ C 3 etc. The A s are increasing so ( ) m (C1 \ C i ) = lim m(c 1 \ C n ) = m(c 1 ) lim m(c n) n n by the preceding proposition. Since m(c 1 ) < we have m(c 1 \ C n ) = m(c 1 ) m(c n ). Also n (C 1 \ C n ) = C 1 \ ( n C n). So ( ) ( ) m (C 1 \ C n ) = m(c 1 ) m Cn = m(c 1 ) lim m(c n). n n Subtracting m(c 1 ) from both sides of the last equation gives the equality in the proposition.

67 Constantin Carathéodory Born: 13 Sept 1873 in Berlin, Germany Died: 2 Feb 1950 in Munich, Germany

68 Axiomatic approach. We will now take the items in Theorem 6 as axioms: Let X be a set. (Usually X will be a topological space or even a metric space). A collection F of subsets of X is called a σ field if: X F, If E F then E c = X \ E F, and If {E n } is a sequence of elements in F then n E n F, The intersection of any family of σ-fields is again a σ-field, and hence given any collection C of subsets of X, there is a smallest σ-field F which contains it. Then F is called the σ-field generated by C.

69 Borel sets. If X is a metric space, the σ-field generated by the collection of open sets is called the Borel σ-field, usually denoted by B or B(X ) and a set belonging to B is called a Borel set.

70 Measures on a σ-field. Given a σ-field F a (non-negative) measure is a function such that m( ) = 0 and m : F [0, ] Countable additivity: If F n is a disjoint collection of sets in F then ( ) m F n = m(f n ). n n In the countable additivity condition it is understood that both sides might be infinite.

71 Outer measures. An outer measure on a set X is a map m to [0, ] defined on the collection of all subsets of X which satisfies m( ) = 0, Monotonicity: If A B then m (A) m (B), and Countable subadditivity: m ( n A n) n m (A n ).

72 Measures from outer measures via Caratheordory. Given an outer measure, m, we defined a set E to be measurable (relative to m ) if m (A) = m (A E) + m (A E c ) for all sets A. Then Caratheodory s theorem that we proved just now asserts that the collection of measurable sets is a σ-field, and the restriction of m to the collection of measurable sets is a measure which we shall usually denote by m.

73 Terminological conflict. There is an unfortunate disagreement in terminology, in that many of the professionals, especially in geometric measure theory, use the term measure for what we have been calling outer measure. However we will follow the above conventions which used to be the old fashioned standard.

74 The next task. An obvious task, given Caratheodory s theorem, is to look for ways of constructing outer measures. This will be the subject of the next lecture.

Solutions to Tutorial 2 (Week 3)

Solutions to Tutorial 2 (Week 3) THE UIVERSITY OF SYDEY SCHOOL OF MATHEMATICS AD STATISTICS Solutions to Tutorial 2 (Week 3) MATH3969: Measure Theory and Fourier Analysis (Advanced) Semester 2, 2016 Web Page: http://sydney.edu.au/science/maths/u/ug/sm/math3969/

More information

Structure of Measurable Sets

Structure of Measurable Sets Structure of Measurable Sets In these notes we discuss the structure of Lebesgue measurable subsets of R from several different points of view. Along the way, we will see several alternative characterizations

More information

NOTES ON MEASURE THEORY. M. Papadimitrakis Department of Mathematics University of Crete. Autumn of 2004

NOTES ON MEASURE THEORY. M. Papadimitrakis Department of Mathematics University of Crete. Autumn of 2004 NOTES ON MEASURE THEORY M. Papadimitrakis Department of Mathematics University of Crete Autumn of 2004 2 Contents 1 σ-algebras 7 1.1 σ-algebras............................... 7 1.2 Generated σ-algebras.........................

More information

REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE

REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE CHRISTOPHER HEIL 1.4.1 Introduction We will expand on Section 1.4 of Folland s text, which covers abstract outer measures also called exterior measures).

More information

Our goal first will be to define a product measure on A 1 A 2.

Our goal first will be to define a product measure on A 1 A 2. 1. Tensor product of measures and Fubini theorem. Let (A j, Ω j, µ j ), j = 1, 2, be two measure spaces. Recall that the new σ -algebra A 1 A 2 with the unit element is the σ -algebra generated by the

More information

Problem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS

Problem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS Problem Set Problem Set #2 Math 5322, Fall 2001 December 3, 2001 ANSWERS i Problem 1. [Problem 18, page 32] Let A P(X) be an algebra, A σ the collection of countable unions of sets in A, and A σδ the collection

More information

Exercise 1. Let E be a given set. Prove that the following statements are equivalent.

Exercise 1. Let E be a given set. Prove that the following statements are equivalent. Real Variables, Fall 2014 Problem set 1 Solution suggestions Exercise 1. Let E be a given set. Prove that the following statements are equivalent. (i) E is measurable. (ii) Given ε > 0, there exists an

More information

If the sets {A i } are pairwise disjoint, then µ( i=1 A i) =

If the sets {A i } are pairwise disjoint, then µ( i=1 A i) = Note: A standing homework assignment for students in MAT1501 is: let me know about any mistakes, misprints, ambiguities etc that you find in these notes. 1. measures and measurable sets If X is a set,

More information

MAA6616 COURSE NOTES FALL 2012

MAA6616 COURSE NOTES FALL 2012 MAA6616 COURSE NOTES FALL 2012 1. σ-algebras Let X be a set, and let 2 X denote the set of all subsets of X. We write E c for the complement of E in X, and for E, F X, write E \ F = E F c. Let E F denote

More information

REAL ANALYSIS I HOMEWORK 2

REAL ANALYSIS I HOMEWORK 2 REAL ANALYSIS I HOMEWORK 2 CİHAN BAHRAN The questions are from Stein and Shakarchi s text, Chapter 1. 1. Prove that the Cantor set C constructed in the text is totally disconnected and perfect. In other

More information

1. Sigma algebras. Let A be a collection of subsets of some fixed set Ω. It is called a σ -algebra with the unit element Ω if. lim sup. j E j = E.

1. Sigma algebras. Let A be a collection of subsets of some fixed set Ω. It is called a σ -algebra with the unit element Ω if. lim sup. j E j = E. Real Analysis, course outline Denis Labutin 1 Measure theory I 1. Sigma algebras. Let A be a collection of subsets of some fixed set Ω. It is called a σ -algebra with the unit element Ω if (a), Ω A ; (b)

More information

Ri and. i=1. S i N. and. R R i

Ri and. i=1. S i N. and. R R i The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

More information

x if x 0, x if x < 0.

x if x 0, x if x < 0. Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

More information

Chap2: The Real Number System (See Royden pp40)

Chap2: The Real Number System (See Royden pp40) Chap2: The Real Number System (See Royden pp40) 1 Open and Closed Sets of Real Numbers The simplest sets of real numbers are the intervals. We define the open interval (a, b) to be the set (a, b) = {x

More information

CHAPTER 5. Product Measures

CHAPTER 5. Product Measures 54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue

More information

MATH41112/61112 Ergodic Theory Lecture Measure spaces

MATH41112/61112 Ergodic Theory Lecture Measure spaces 8. Measure spaces 8.1 Background In Lecture 1 we remarked that ergodic theory is the study of the qualitative distributional properties of typical orbits of a dynamical system and that these properties

More information

Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year , First Semester Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

More information

x = (x 1,..., x d ), y = (y 1,..., y d ) x + y = (x 1 + y 1,..., x d + y d ), x = (x 1,..., x d ), t R tx = (tx 1,..., tx d ).

x = (x 1,..., x d ), y = (y 1,..., y d ) x + y = (x 1 + y 1,..., x d + y d ), x = (x 1,..., x d ), t R tx = (tx 1,..., tx d ). 1. Lebesgue Measure 2. Outer Measure 3. Radon Measures on R 4. Measurable Functions 5. Integration 6. Coin-Tossing Measure 7. Product Measures 8. Change of Coordinates 1. Lebesgue Measure Throughout R

More information

Chapter 1. Sigma-Algebras. 1.1 Definition

Chapter 1. Sigma-Algebras. 1.1 Definition Chapter 1 Sigma-Algebras 1.1 Definition Consider a set X. A σ algebra F of subsets of X is a collection F of subsets of X satisfying the following conditions: (a) F (b) if B F then its complement B c is

More information

Math 563 Measure Theory Project 1 (Funky Functions Group) Luis Zerón, Sergey Dyachenko

Math 563 Measure Theory Project 1 (Funky Functions Group) Luis Zerón, Sergey Dyachenko Math 563 Measure Theory Project (Funky Functions Group) Luis Zerón, Sergey Dyachenko 34 Let C and C be any two Cantor sets (constructed in Exercise 3) Show that there exists a function F: [,] [,] with

More information

1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. 1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

More information

Analysis I. 1 Measure theory. Piotr Haj lasz. 1.1 σ-algebra.

Analysis I. 1 Measure theory. Piotr Haj lasz. 1.1 σ-algebra. Analysis I Piotr Haj lasz 1 Measure theory 1.1 σ-algebra. Definition. Let be a set. A collection M of subsets of is σ-algebra if M has the following properties 1. M; 2. A M = \ A M; 3. A 1, A 2, A 3,...

More information

The Lebesgue Measure and Integral

The Lebesgue Measure and Integral The Lebesgue Measure and Integral Mike Klaas April 12, 2003 Introduction The Riemann integral, defined as the limit of upper and lower sums, is the first example of an integral, and still holds the honour

More information

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

More information

Limits and convergence.

Limits and convergence. Chapter 2 Limits and convergence. 2.1 Limit points of a set of real numbers 2.1.1 Limit points of a set. DEFINITION: A point x R is a limit point of a set E R if for all ε > 0 the set (x ε,x + ε) E is

More information

Math 507/420 Homework Assignment #1: Due in class on Friday, September 20. SOLUTIONS

Math 507/420 Homework Assignment #1: Due in class on Friday, September 20. SOLUTIONS Math 507/420 Homework Assignment #1: Due in class on Friday, September 20. SOLUTIONS 1. Show that a nonempty collection A of subsets is an algebra iff 1) for all A, B A, A B A and 2) for all A A, A c A.

More information

Course 421: Algebraic Topology Section 1: Topological Spaces

Course 421: Algebraic Topology Section 1: Topological Spaces Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

More information

) + ˆf (n) sin( 2πnt. = 2 u x 2, t > 0, 0 < x < 1. u(0, t) = u(1, t) = 0, t 0. (x, 0) = 0 0 < x < 1.

) + ˆf (n) sin( 2πnt. = 2 u x 2, t > 0, 0 < x < 1. u(0, t) = u(1, t) = 0, t 0. (x, 0) = 0 0 < x < 1. Introduction to Fourier analysis This semester, we re going to study various aspects of Fourier analysis. In particular, we ll spend some time reviewing and strengthening the results from Math 425 on Fourier

More information

Extension of measure

Extension of measure 1 Extension of measure Sayan Mukherjee Dynkin s π λ theorem We will soon need to define probability measures on infinite and possible uncountable sets, like the power set of the naturals. This is hard.

More information

POSITIVE INTEGERS, INTEGERS AND RATIONAL NUMBERS OBTAINED FROM THE AXIOMS OF THE REAL NUMBER SYSTEM

POSITIVE INTEGERS, INTEGERS AND RATIONAL NUMBERS OBTAINED FROM THE AXIOMS OF THE REAL NUMBER SYSTEM MAT 1011 TECHNICAL ENGLISH I 03.11.2016 Dokuz Eylül University Faculty of Science Department of Mathematics Instructor: Engin Mermut Course assistant: Zübeyir Türkoğlu web: http://kisi.deu.edu.tr/engin.mermut/

More information

Math Real Analysis I

Math Real Analysis I Math 431 - Real Analysis I Solutions to Homework due October 1 In class, we learned of the concept of an open cover of a set S R n as a collection F of open sets such that S A. A F We used this concept

More information

6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, ) 6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

More information

Sequences and Convergence in Metric Spaces

Sequences and Convergence in Metric Spaces Sequences and Convergence in Metric Spaces Definition: A sequence in a set X (a sequence of elements of X) is a function s : N X. We usually denote s(n) by s n, called the n-th term of s, and write {s

More information

God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886)

God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886) Chapter 2 Numbers God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886) God created the integers and the rest is the work

More information

General theory of stochastic processes

General theory of stochastic processes CHAPTER 1 General theory of stochastic processes 1.1. Definition of stochastic process First let us recall the definition of a random variable. A random variable is a random number appearing as a result

More information

and s n (x) f(x) for all x and s.t. s n is measurable if f is. REAL ANALYSIS Measures. A (positive) measure on a measurable space

and s n (x) f(x) for all x and s.t. s n is measurable if f is. REAL ANALYSIS Measures. A (positive) measure on a measurable space RAL ANALYSIS A survey of MA 641-643, UAB 1999-2000 M. Griesemer Throughout these notes m denotes Lebesgue measure. 1. Abstract Integration σ-algebras. A σ-algebra in X is a non-empty collection of subsets

More information

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

More information

SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

More information

MEASURE AND INTEGRATION. Dietmar A. Salamon ETH Zürich

MEASURE AND INTEGRATION. Dietmar A. Salamon ETH Zürich MEASURE AND INTEGRATION Dietmar A. Salamon ETH Zürich 12 May 2016 ii Preface This book is based on notes for the lecture course Measure and Integration held at ETH Zürich in the spring semester 2014. Prerequisites

More information

Math 317 HW #7 Solutions

Math 317 HW #7 Solutions Math 17 HW #7 Solutions 1. Exercise..5. Decide which of the following sets are compact. For those that are not compact, show how Definition..1 breaks down. In other words, give an example of a sequence

More information

Duality of linear conic problems

Duality of linear conic problems Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

More information

Course 214 Section 1: Basic Theorems of Complex Analysis Second Semester 2008

Course 214 Section 1: Basic Theorems of Complex Analysis Second Semester 2008 Course 214 Section 1: Basic Theorems of Complex Analysis Second Semester 2008 David R. Wilkins Copyright c David R. Wilkins 1989 2008 Contents 1 Basic Theorems of Complex Analysis 1 1.1 The Complex Plane........................

More information

3 σ-algebras. Solutions to Problems

3 σ-algebras. Solutions to Problems 3 σ-algebras. Solutions to Problems 3.1 3.12 Problem 3.1 (i It is clearly enough to show that A, B A = A B A, because the case of N sets follows from this by induction, the induction step being A 1...

More information

MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I. A brief introduction to proofs, sets, and functions MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

More information

POWER SETS AND RELATIONS

POWER SETS AND RELATIONS POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

More information

Probability Foundations for Electrical Engineers Prof. Krishna Jagannathan Department of Electrical Engineering Indian Institute of Technology, Madras

Probability Foundations for Electrical Engineers Prof. Krishna Jagannathan Department of Electrical Engineering Indian Institute of Technology, Madras Probability Foundations for Electrical Engineers Prof. Krishna Jagannathan Department of Electrical Engineering Indian Institute of Technology, Madras Lecture - 9 Borel Sets and Lebesgue Measure -1 (Refer

More information

Lebesgue Measure on R n

Lebesgue Measure on R n 8 CHAPTER 2 Lebesgue Measure on R n Our goal is to construct a notion of the volume, or Lebesgue measure, of rather general subsets of R n that reduces to the usual volume of elementary geometrical sets

More information

Metric Spaces. Chapter 1

Metric Spaces. Chapter 1 Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

More information

Notes 2 for Honors Probability and Statistics

Notes 2 for Honors Probability and Statistics Notes 2 for Honors Probability and Statistics Ernie Croot August 24, 2010 1 Examples of σ-algebras and Probability Measures So far, the only examples of σ-algebras we have seen are ones where the sample

More information

x a x 2 (1 + x 2 ) n.

x a x 2 (1 + x 2 ) n. Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number

More information

14 Abian and Kemp Theorem 1. For every cardinal k if the union of k many sets of measure zero is of measure zero, then the union of k many measurable

14 Abian and Kemp Theorem 1. For every cardinal k if the union of k many sets of measure zero is of measure zero, then the union of k many measurable PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE Nouvelle série tome 52 (66), 1992, 13 17 ON MEASURABILITY OF UNCOUNTABLE UNIONS OF MEASURABLE SETS Alexander Abian and Paula Kemp Abstract. This paper deals exclusively

More information

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

More information

Lecture Notes on Measure Theory and Functional Analysis

Lecture Notes on Measure Theory and Functional Analysis Lecture Notes on Measure Theory and Functional Analysis P. Cannarsa & T. D Aprile Dipartimento di Matematica Università di Roma Tor Vergata cannarsa@mat.uniroma2.it daprile@mat.uniroma2.it aa 2006/07 Contents

More information

Chapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1

Chapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1 Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes

More information

Prof. Girardi, Math 703, Fall 2012 Homework Solutions: Homework 13. in X is Cauchy.

Prof. Girardi, Math 703, Fall 2012 Homework Solutions: Homework 13. in X is Cauchy. Homework 13 Let (X, d) and (Y, ρ) be metric spaces. Consider a function f : X Y. 13.1. Prove or give a counterexample. f preserves convergent sequences if {x n } n=1 is a convergent sequence in X then

More information

Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

More information

STAT 571 Assignment 1 solutions

STAT 571 Assignment 1 solutions STAT 571 Assignment 1 solutions 1. If Ω is a set and C a collection of subsets of Ω, let A be the intersection of all σ-algebras that contain C. rove that A is the σ-algebra generated by C. Solution: Let

More information

Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.

Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent. MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we

More information

Proof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.

Proof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems. Math 232 - Discrete Math 2.1 Direct Proofs and Counterexamples Notes Axiom: Proposition that is assumed to be true. Proof: A logical argument establishing the truth of the theorem given the truth of the

More information

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

More information

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

More information

Math 317 HW #5 Solutions

Math 317 HW #5 Solutions Math 317 HW #5 Solutions 1. Exercise 2.4.2. (a) Prove that the sequence defined by x 1 = 3 and converges. x n+1 = 1 4 x n Proof. I intend to use the Monotone Convergence Theorem, so my goal is to show

More information

Class Notes for MATH 355.

Class Notes for MATH 355. Class Notes for MATH 355. by S. W. Drury Copyright c 2002 2007, by S. W. Drury. Contents Starred chapters and sections should be omitted on a first reading. Double starred sections should be omitted on

More information

LEARNING OBJECTIVES FOR THIS CHAPTER

LEARNING OBJECTIVES FOR THIS CHAPTER CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional

More information

CHAPTER 3. Sequences. 1. Basic Properties

CHAPTER 3. Sequences. 1. Basic Properties CHAPTER 3 Sequences We begin our study of analysis with sequences. There are several reasons for starting here. First, sequences are the simplest way to introduce limits, the central idea of calculus.

More information

Infinite product spaces

Infinite product spaces Chapter 7 Infinite product spaces So far we have talked a lot about infinite sequences of independent random variables but we have never shown how to construct such sequences within the framework of probability/measure

More information

Compactness in metric spaces

Compactness in metric spaces MATHEMATICS 3103 (Functional Analysis) YEAR 2012 2013, TERM 2 HANDOUT #2: COMPACTNESS OF METRIC SPACES Compactness in metric spaces The closed intervals [a, b] of the real line, and more generally the

More information

Elements of probability theory

Elements of probability theory 2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted

More information

MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

More information

(Refer Slide Time: 1:41)

(Refer Slide Time: 1:41) Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 10 Sets Today we shall learn about sets. You must

More information

Open and Closed Sets

Open and Closed Sets Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.

More information

10.2 Series and Convergence

10.2 Series and Convergence 10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and

More information

1 Limiting distribution for a Markov chain

1 Limiting distribution for a Markov chain Copyright c 2009 by Karl Sigman Limiting distribution for a Markov chain In these Lecture Notes, we shall study the limiting behavior of Markov chains as time n In particular, under suitable easy-to-check

More information

This chapter describes set theory, a mathematical theory that underlies all of modern mathematics.

This chapter describes set theory, a mathematical theory that underlies all of modern mathematics. Appendix A Set Theory This chapter describes set theory, a mathematical theory that underlies all of modern mathematics. A.1 Basic Definitions Definition A.1.1. A set is an unordered collection of elements.

More information

Polish spaces and standard Borel spaces

Polish spaces and standard Borel spaces APPENDIX A Polish spaces and standard Borel spaces We present here the basic theory of Polish spaces and standard Borel spaces. Standard references for this material are the books [143, 231]. A.1. Polish

More information

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem

More information

Measure Theory. 1 Measurable Spaces

Measure Theory. 1 Measurable Spaces 1 Measurable Spaces Measure Theory A measurable space is a set S, together with a nonempty collection, S, of subsets of S, satisfying the following two conditions: 1. For any A, B in the collection S,

More information

CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

More information

P (A) = lim P (A) = N(A)/N,

P (A) = lim P (A) = N(A)/N, 1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or non-deterministic experiments. Suppose an experiment can be repeated any number of times, so that we

More information

Lecture 2 : Basics of Probability Theory

Lecture 2 : Basics of Probability Theory Lecture 2 : Basics of Probability Theory When an experiment is performed, the realization of the experiment is an outcome in the sample space. If the experiment is performed a number of times, different

More information

NONMEASURABLE ALGEBRAIC SUMS OF SETS OF REALS

NONMEASURABLE ALGEBRAIC SUMS OF SETS OF REALS NONMEASURABLE ALGEBRAIC SUMS OF SETS OF REALS MARCIN KYSIAK Abstract. We present a theorem which generalizes some known theorems on the existence of nonmeasurable (in various senses) sets of the form X+Y.

More information

Measuring sets with translation invariant Borel measures

Measuring sets with translation invariant Borel measures Measuring sets with translation invariant Borel measures University of Warwick Fractals and Related Fields III, 21 September 2015 Measured sets Definition A Borel set A R is called measured if there is

More information

Topology and Convergence by: Daniel Glasscock, May 2012

Topology and Convergence by: Daniel Glasscock, May 2012 Topology and Convergence by: Daniel Glasscock, May 2012 These notes grew out of a talk I gave at The Ohio State University. The primary reference is [1]. A possible error in the proof of Theorem 1 in [1]

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

Notes on metric spaces

Notes on metric spaces Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

More information

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify

More information

The Alternating Series Test

The Alternating Series Test The Alternating Series Test So far we have considered mostly series all of whose terms are positive. If the signs of the terms alternate, then testing convergence is a much simpler matter. On the other

More information

3. Recurrence Recursive Definitions. To construct a recursively defined function:

3. Recurrence Recursive Definitions. To construct a recursively defined function: 3. RECURRENCE 10 3. Recurrence 3.1. Recursive Definitions. To construct a recursively defined function: 1. Initial Condition(s) (or basis): Prescribe initial value(s) of the function.. Recursion: Use a

More information

Metric Spaces Joseph Muscat 2003 (Last revised May 2009)

Metric Spaces Joseph Muscat 2003 (Last revised May 2009) 1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of

More information

Assignment 7; Due Friday, November 11

Assignment 7; Due Friday, November 11 Assignment 7; Due Friday, November 9.8 a The set Q is not connected because we can write it as a union of two nonempty disjoint open sets, for instance U = (, 2) and V = ( 2, ). The connected subsets are

More information

Introduction to Topology

Introduction to Topology Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................

More information

GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G. Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

More information

Lecture 12: Sequences. 4. If E X and x is a limit point of E then there is a sequence (x n ) in E converging to x.

Lecture 12: Sequences. 4. If E X and x is a limit point of E then there is a sequence (x n ) in E converging to x. Lecture 12: Sequences We now recall some basic properties of limits. Theorem 0.1 (Rudin, Theorem 3.2). Let (x n ) be a sequence in a metric space X. 1. (x n ) converges to x X if and only if every neighborhood

More information

1. R In this and the next section we are going to study the properties of sequences of real numbers.

1. R In this and the next section we are going to study the properties of sequences of real numbers. +a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

More information

Practice Problems Solutions

Practice Problems Solutions Practice Problems Solutions The problems below have been carefully selected to illustrate common situations and the techniques and tricks to deal with these. Try to master them all; it is well worth it!

More information

Math 230a HW3 Extra Credit

Math 230a HW3 Extra Credit Math 230a HW3 Extra Credit Erik Lewis November 15, 2006 1 Extra Credit 16. Regard Q, the set of all rational numbers, as a metric space, with d(p, q) = p q. Let E be the set of all p Q such that 2 < p

More information

Mathematics for Econometrics, Fourth Edition

Mathematics for Econometrics, Fourth Edition Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents

More information

HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)! Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

More information

MATHEMATICS 117/E-217, SPRING 2012 PROBABILITY AND RANDOM PROCESSES WITH ECONOMIC APPLICATIONS Module #5 (Borel Field and Measurable Functions )

MATHEMATICS 117/E-217, SPRING 2012 PROBABILITY AND RANDOM PROCESSES WITH ECONOMIC APPLICATIONS Module #5 (Borel Field and Measurable Functions ) MATHEMATICS 117/E-217, SPRING 2012 PROBABILITY AND RANDOM PROCESSES WITH ECONOMIC APPLICATIONS Module #5 (Borel Field and Measurable Functions ) Last modified: May 1, 2012 Reading Dineen, pages 57-67 Proof

More information